Question

The two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is $60^{\circ}$. If the area of the quadrilateral is $4 \sqrt{3}$ , then the perimeter of the quadrilateral is :

• A. 12.5
• B. 13.2
• C. 12
• D. 13

Answer 1

Answer: (B)

$\cos 60=\frac{4+25-C^{2}}{2.2.5}$
$10=29-C^{2}$
$C^{2}=19$
$C=\sqrt{19}$
$-\frac{1}{2}=\frac{a^{2}+b^{2}-19}{2 a b}$
$a^{2}+b^{2}-19=-a b$
$a^{2}+b^{2}+a b=19$
Area $=\frac{1}{2} \times 2 \times 5 \sin 60+\frac{1}{2} a b \sin x=4 \sqrt{3}$
$\frac{5 \sqrt{3}}{2}+\frac{a b \sqrt{3}}{4}=4 \sqrt{3}$
$\frac{a b}{4}=4-\frac{5}{2}=\frac{3}{2}$
$a b=6$
$a^{2}+b^{2}=13$
$a=2, b=3$
Perimeter
$=2+5+2+3$
$=12$